On nonsingularity of circulant matrices
Abstract
In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2k+1$ is either a power of a prime, or a product of two distinct primes. For any other integer $2k+1$ we construct circulant matrices having determinant $0$. The smallest singular matrix appears when $2k+1=45$. The possibility for such matrices to be singular is rather low, smaller than $10^{-4}$ in this case.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2018
- DOI:
- 10.48550/arXiv.1810.09893
- arXiv:
- arXiv:1810.09893
- Bibcode:
- 2018arXiv181009893C
- Keywords:
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- Mathematics - Commutative Algebra;
- 15B05;
- 11R18;
- 68P30;
- 94A05
- E-Print:
- 12 pages. To be published in Linear Algebra and Its Applications